Increase Our Learning Horizon wih Evolving Technology Wei-Chi Yang Radford Universiy Radford, Virginia 24142, USA e-mail: wyang@radford.edu Asrac. In his shor noe, we demonsrae how evolving echnological ools can allow us o enhance our learning horizon wherever learners levels are in mahemaics. We will use examples o show Dynamic Geomery DG and is animaion and numerical approximaions can make mahemaics accessile o more learners. In he meanime, prolems can e made challenging when learners aemp o generalize he resul y using a Compuer Algera Sysem CAS. Finally, we show some resuls valid in wo dimensions can e exended o hree dimensional ones oo. 1. Mahemaics conens can e made accessile and challenging The parameric graph of a curve is an imporan concep when learners expand heir knowledge of he graph of a funcion. The example elow is o ask when a circle is drawn, wha could we ge ou of he parameric equaions. Example 1. A circle is drawn arirarily using a Dynamic Geomery Sysem, which we use ClassPad see [1] for demonsraion. Then pick a poin P=x,y on he circle, where is ranging from 0 o 2 π. Wha are he graphs of, x and, y respecively. We sar wih he following circle, which has cener C= [-1.35,-0.225] and radius r=1.179248.
Figure 1. We pick a poin P=x,y and le i animae once on he circle and oained he following screen shos: Figures 2-7. In he mean ime, we collec he,x and,y respecively and we show he parial lis as follows: Figure 8..
Finally, we drag he columns Direcion which is he angle and x ack o he Figure 1, we see he following curve in addiion o he circle. Can you name ha curve? Figure 9. Similarly, we drag he columns Direcion which is he angle and y ack o he Figure 1, we see he following curve in addiion o he circle. Can you name ha curve? Figure 10. Readers can link o a video clip which demonsraes he processes aove see [2]. The aove exercise is o moivae sudens o explore he parameric equaion of he form [x,y]=[a+r*cos,+r*sin], where a, is he cener of he circle of radius r. Experienced learners know he aove parameric equaion represen a circle. However, when we pose he quesion reversely, he prolem ecomes more ineresing. 2. Inegraing Dynamic Geomery wih CAS W may use he numerical capailiy of DG o approximae a soluion. This makes mahemaics accessile o more sudens efore one ges ino inroducing he heory ehind a concep. So his ype of proem can e inroduced o learners wih minimum knowledge. In he mean ime, he prolem can e re-inroduced again and ecomes challenging when we use a CAS o verify if our conjecure is rue. This spiral way of learning process
makes sudens undersand he connecion eween he old knowledge hey learned from he pas and make a good argumen why new knowlege needs o e inroduced. Example 2. We are given a line L of he form y = mx + and a curve S see Figure 11 elow, find he reflecion of he curve S respecive o he line L. Figure 11. Assume he general case for finding he inverse of [ x, y ] wih respec o y = mx +. We firs se θ = an 1 m. We call h e reflecion of [ x, y ] wih respec o y = mx + o e [ p, q ]. I can e proved ha p q cos2 sin2 sin2 cos2 x y 0. Example 3. 2cos cos2 Find he reflecion of 2sin sin2 wih respec o y 2x 1. We se p θ = an 2 and we oain q o e.,
We plo hick ogeher wih hin elow: q p y x Figure 12. Example 4. Hypocyloid Find he reflecion of 1 sin 1 cos a a a 3 + + sin cos a = a and = 1 q p wih respec o We se and ain o e 2 +1. = x y 2 an 1 = θ we o We plo hick ogeher wih hin elow: q p y x
. Figure 13. 3. Going from 2D o 3D The example elow shows how we can link he idea of Lagrange Mulipliers, is geomeric inerpreaion wih linear indecency of Linear Algera ogeher wih he help of Dynamic Geomery Sofware packages. Example 5. We are given hree curves in he plane, see C 1, 2 and elow. We need o find poins A, B, and C on C 1, C2 and C 3 respecively so ha he disance AB + AC achieves is minimum. C C 3 Figure 14. We noe ha we may use a DG sofware o make conjecures of when he minimum disance should occur, u wih some knowledge of Linear Algera and Muli-variale Calculus, i is no hard o make he following oservaions:
AB should e parallel o he normal vecor of he curve C2 a B. AC should e parallel o he normal vecor of he curve C3 a C. We shoul d place h e poins A, B and C on C1, C2 and C3 respecively so ha he normal vecor of C1 a A=linear cominaion of AB and AC. The aove oservaion is precisely wha we expec from applying echnique of Lagrange Mulipliers. We exend he concep o 3D where we can use a 3D DG such as Cari 3D o explore. Example 6. We are given four surfaces in he space, represened y he orange surface, called S1; yellow surface, called S 2 ; lue surface called S 3 and he purple surface, called S 4 respecively. We wan o find poins A, B,C and D on S1, S2, S3 and S 4 respecively so ha he disance AB + AC + AD achieves is minimum. Figure 15. I is naural o have he following oservaions: AB should e parallel o he normal vecor of he surface S2 a B. AC should e parallel o he normal vecor of he surface S3 a C. AD should e parallel o he normal vecor of he surface S4 a D. We should place he poins A, B, C and D on S1, S2, S3 and S4 respecively so ha The normal vecor of S1 a A=linear cominaion of AB, AC and AD.
Cerainly he aove concep can e exended o any finie dimensions oo. 4. Conclusion Technology definiely can no solves all our prolems. Implemening echnological ools ino eaching and learning is a no rivial ask and i will e an on-going discussion issue many years o come. One of he issues ha we heard ofen is ha many sudens have los confidence or ineress efore enering universiies ecause of he deficency in algeraic manipulaion skills. I may e possile o uild a curriculum around mahemaical programs when eachers inroduces DG so mahemaics is more accessile o sudens a younger age and inspire sudens why Algera is needed. If mahemaics prolems are chosen properly, same prolems hey encounered in he pas can e re-solved again using heir added knowledge when sudens enering universiies. From he examples we see aove, DG indeed can make mahemaics more accessile, and when a CAS is used o prove resuls analyically, mahemaics is challenging oo. Acknowledgemen. Auhor would like o hank Jean-Jacques Dahan of France of creaing he Cari 3D see [3] file for Figure 15. References [1] ClassPad, a produc of CASIO Compuer LTd, hp://classpad.ne or hp://classpad.org. [2] A video clip, hp://mahandech.org/casio_video/trig_shifing2/trig_shifing2.hml. [3] Cari 3D, a produc of Carilog, hp://cari.com.